Let $i(G)$ be the number of isolated vertices in a graph $G$. The isolated toughness of $G$ is defined as $I(G)=\infty $ if $G$ is complete, and $I(G)=\operatorname{min}\{|S|/i(G-S) : S\subseteq V(G),\, i(G-S)\ge 2\}$ otherwise. We show that $G$ is a frac

A graph $G$ is called a fractional $(g,f,n’,m)$-critical deleted graph if it remains a fractional $(g,f,m)$-deleted graph after deleting any $n’$ vertices. We prove that if $G$ is a graph of order $n$, $1\le a\le g(x)\le f(x)\le b$ for any $x\in V(G)$, $\

Graphs are used to model chemical compounds and drugs. In the graphs, each vertex represents an atom of molecule and edges between the corresponding vertices are used to represent covalent bounds between atoms. We call such a graph, which is derived from a chemical compound, a molecular graph. Evidence shows that the vertex-weighted Wiener number, which is defined over this molecular graph, is strongly correlated to both the melting point and boiling point of the compounds. In this paper, we report the extremal vertex-weighted Wiener number of bicyclic molecular graph in terms of molecular structural analysis and graph transformations. The promising prospects of the application for the chemical and pharmacy engineering are illustrated by theoretical results achieved in this paper.